How successfully can convnets detect NSFW images?
The 2015 paper entitled "Applying deep learning to classify pornographic images and videos" applied various types of convnets for detecting pornography. The proposed architecture achieved 94.1% accuracy on the NPDI dataset, which contains 800 videos (400 porn, 200 non-porn "easy" and 200 non-porn "difficult"). More traditional computer vision methods achieved 90.9% accuracy. The proposed architecture also performs very well regarding the ROC curve.

There does not seem to exist any works regarding the other aspects of NSFW yet.